Kite and its Theorems

In this section, we will discuss kite and its theorems.
In kite, adjacent sides are equal and long diagonal bisect the small diagonal at right angle.All interior angles are acute angles.Theorem 1 : If a quadrilateral is a kite, then its diagonals are perpendicular.

GIVEN : AB ≅ CB and AD ≅ CD

PROVE THAT : AC ⊥ BD

Proof :

Statements

Reasons

1)AB ≅ AD

1) Given

2) BC ≅ CD

2) Given

3) AC ≅ AC

3) Reflexive (common side)

4) ΔABC ≅ ΔADC

4) SSS Postulates

5) ∠BAE ≅ ∠DAE

5) CPCTC

6) ΔABD is an Isosceles triangle.

6) By property of an isosceles triangle.

7) ∠ABE ≅ ∠ADE

7) Property of isosceles triangle.

8) ΔABE ≅ ΔADE

8) ASA postulate.

9) ∠AEB ≅ ∠AED

9) CPCTC

10)∠AEB +∠AED = 180

10) Linear pair angles are supplementary.

11) 2∠AEB = 180

11) Addition property

12) ∠AEB = 90

12) Division property

13) AC ⊥ BD and AE ⊥ BD

13) By property of perpendicular.

Theorem 2: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.